How to make an object go through a plane but the plane deform with the face of the object?
One way to achieve this is with Cloth physics.
Start by animating your object raising up through a flat plane (keyframe it starting below the surface and then 'rise up' until it protrudes). Add physics Collision to both the plane and your object.
Create another plane and position this above the first. Heavily subdivide it so that the cloth can suitably drape over your object and add Cloth physics to it.
Run the animation and you should have something like the following :

(note that I artifically reversed the same frames to go back to the start of the animation - that isn't possible with the cloth physics!!)
You might want to tweak the cloth physics paramters to get the effect you're looking for.